Answer
Verified
406k+ views
Hint: When the sphere just touches the inner surface of the cone, then the surface of the cone is nothing but a tangent to the sphere. Also, the centre of the sphere is R distance below the surface of the base of the cone (Where R is the radius of the sphere)
The volume of a sphere is
\[=\dfrac{4}{3}\pi {{r}^{3}}\]
(Where ‘r ‘ is the radius of the sphere)
The volume of a cone is
\[=\dfrac{1}{3}\pi {{r}^{2}}h\]
(Where ‘r’ is the radius of the cone and’ h’ is the height of the cone)
Complete step-by-step answer:
As mentioned in the question, the sphere is completely immersed in the cone.
Now, by referring to the figure, we can say that
\[\begin{align}
& \angle CAB={{\tan }^{-1}}\dfrac{8}{6} \\
& \angle CAB={{\tan }^{-1}}\dfrac{4}{3} \\
& \angle CAB={{53}^{\circ }} \\
\end{align}\]
Now, if
\[\begin{align}
& \angle CAB={{53}^{\circ }} \\
& \therefore \angle BCA={{37}^{\circ }} \\
\end{align}\]
Now, on applying sin formula in \[\vartriangle PCO\] , we get
\[\begin{align}
& \sin {{37}^{\circ }}=\dfrac{R}{\left( 8-R \right)} \\
& \dfrac{3}{5}=\dfrac{R}{\left( 8-R \right)} \\
& 24-3R=5R \\
& R=3 \\
\end{align}\]
(Where R is the radius of the sphere)
Therefore, using the formula for finding the volume of a sphere as it is given in the hint, we get
\[\begin{align}
& =\dfrac{4}{3}\pi {{(3)}^{3}} \\
& =4\times 9\pi \\
& =36\pi \\
\end{align}\]
Now, the fraction of water which overflows is given as follows
\[\begin{align}
& =\dfrac{36\pi }{\dfrac{1}{3}\pi {{6}^{2}}\times 8} \\
& =\dfrac{3}{8} \\
\end{align}\]
Hence, the fraction of water that overflows is \[\dfrac{3}{8}\] .
Note: The students can make an error if they don’t know the formulas for volume of sphere and cone and also the trigonometric ratio which are given in hint as
The volume of a sphere is
\[=\dfrac{4}{3}\pi {{r}^{3}}\]
(Where ‘r ‘ is the radius of the sphere)
The volume of a cone is
\[=\dfrac{1}{3}\pi {{r}^{2}}h\]
(where ‘r’ is the radius of the cone and’ h’ is the height of the cone)
The volume of a sphere is
\[=\dfrac{4}{3}\pi {{r}^{3}}\]
(Where ‘r ‘ is the radius of the sphere)
The volume of a cone is
\[=\dfrac{1}{3}\pi {{r}^{2}}h\]
(Where ‘r’ is the radius of the cone and’ h’ is the height of the cone)
Complete step-by-step answer:
As mentioned in the question, the sphere is completely immersed in the cone.
Now, by referring to the figure, we can say that
\[\begin{align}
& \angle CAB={{\tan }^{-1}}\dfrac{8}{6} \\
& \angle CAB={{\tan }^{-1}}\dfrac{4}{3} \\
& \angle CAB={{53}^{\circ }} \\
\end{align}\]
Now, if
\[\begin{align}
& \angle CAB={{53}^{\circ }} \\
& \therefore \angle BCA={{37}^{\circ }} \\
\end{align}\]
Now, on applying sin formula in \[\vartriangle PCO\] , we get
\[\begin{align}
& \sin {{37}^{\circ }}=\dfrac{R}{\left( 8-R \right)} \\
& \dfrac{3}{5}=\dfrac{R}{\left( 8-R \right)} \\
& 24-3R=5R \\
& R=3 \\
\end{align}\]
(Where R is the radius of the sphere)
Therefore, using the formula for finding the volume of a sphere as it is given in the hint, we get
\[\begin{align}
& =\dfrac{4}{3}\pi {{(3)}^{3}} \\
& =4\times 9\pi \\
& =36\pi \\
\end{align}\]
Now, the fraction of water which overflows is given as follows
\[\begin{align}
& =\dfrac{36\pi }{\dfrac{1}{3}\pi {{6}^{2}}\times 8} \\
& =\dfrac{3}{8} \\
\end{align}\]
Hence, the fraction of water that overflows is \[\dfrac{3}{8}\] .
Note: The students can make an error if they don’t know the formulas for volume of sphere and cone and also the trigonometric ratio which are given in hint as
The volume of a sphere is
\[=\dfrac{4}{3}\pi {{r}^{3}}\]
(Where ‘r ‘ is the radius of the sphere)
The volume of a cone is
\[=\dfrac{1}{3}\pi {{r}^{2}}h\]
(where ‘r’ is the radius of the cone and’ h’ is the height of the cone)
Recently Updated Pages
Select the antonym for the following word from the class 10 english CBSE
Select the antonym for the following word from the class 10 english CBSE
Select the antonym for the following word from the class 10 english CBSE
Select the antonym for the following word from the class 10 english CBSE
Select the antonym for the following word from the class 10 english CBSE
Select the antonym for the following word from the class 10 english CBSE
Trending doubts
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Change the following sentences into negative and interrogative class 10 english CBSE
Chahalgani means ATurkish noble under Iltutmish BSlaves class 10 social science CBSE
Why is there a time difference of about 5 hours between class 10 social science CBSE
Explain the Treaty of Vienna of 1815 class 10 social science CBSE
the Gond raja of Garha Katanga assumed the title of class 10 social science CBSE